On the duality of complex geometry and material heterogeneities in linear elastodynamics

Wave interaction with curvilinear boundaries is a topic of great significance in many fields including seismology, fracture mechanics, and metamaterials. Here we present a novel procedure for evaluation of the influence of complex geometry on wave propagation in quasi-1D structures, without integration of equations of motion, by using a coordinate transformation technique that maps the stress-free domain with curved boundaries into another stress-free domain with a flat surface. It follows that the elastic tensor and material density transform into modified quantities. Through the analysis of the acoustic tensor in the mapped configuration, we show it is possible to make correlations between curvilinear geometry in the reference configuration and variation in impedance in the mapped configuration. This qualitatively reveals domains of possible amplification and suppression and emergence of band gaps. In particular, we show that both valleys and hills amplify wave motion to varying degrees depending on their steepness. Furthermore, this transformation provides insight into the emergence and evolution of band gaps in curved geometries. Specifically, with increased surface waviness, the contrast in the impedance in the mapped domain increases, enriching the band gap spectrum and leading to the emergence of lower frequency stop bands. This framework opens new opportunities for understanding wave propagation in domains with curved geometries and provides new pathways for achieving extreme properties in homogeneous materials by leveraging curvilinear geometric effects.